for a field range of
\(H_{\textrm {c1}\perp } \ll H_{\textrm {irr}} < H \ll H_{\textrm {c2}\perp }\) 
(\(H_{\textrm {irr}}\) : irreversible magnetic field above which the magnetization is reversible,
\(H_{\textrm {c2}\perp }\) : perpendicular upper critical field,
\(\phi _{0}\) : flux quantum, \(\alpha \) : core contribution factor,
\(\beta \) : a constant of the order of unity) [1]}.
In this field range, vortices are free from collective pinning, and magnetization is
reversible against field sweeps.
According to Eq. (1), \(dM/d(\ln H)\)  gives an estimate of \(\lambda _{\parallel }\) .
As the correction factor \(\alpha \)  from the London model, which does not consider
the vortex core contribution, we employed \(\alpha \)  = 0.77, a value obtained in Ref. [2]}.
The \(M(H)\)  curves at fixed temperatures were measured under upward and downward field sweeps in
the range of 0-5 T.
Fig. REF (a) and REF (b) shows the \(M(H)\)  curve of \(\kappa \) -HgBr at
1.8 K along with that of the half-filled system \(\kappa \) -Br at 5 K
(the same reduced temperature, \(T/T_{\textrm {c}}\sim \) 0.42) for comparison.
\(\kappa \) -HgBr exhibits a peak in \(M(H)\)  at \(H_{\textrm {p}}\)  = 30 Oe and
reversible behavior above \(H_{\textrm {irr}}\)  = 300 Oe;
both the \(H_{\textrm {p}}\)  and \(H_{\textrm {irr}}\)  values are approximately an order of
magnitude below those in \(\kappa \) -Br [3]},
suggesting that the superconductivity in \(\kappa \) -HgBr is highly resilient to magnetic fields.
Accordingly, the temperature dependence of \(H_{\textrm {irr}}\)  shows that
the vortex liquid phase is largely extended in the \(H-T\)  plane in \(\kappa \) -HgBr in comparison with
\(\kappa \) -Br (Fig. REF (c)).
The extended vortex liquid state in superconductivity with BEC-like character is
theoretically suggested [4]}.
Magnetization curves above \(H_{\textrm {irr}}\)  at several temperatures are shown in
Fig. REF (a).
The gray lines indicate the fits of Eq. (1),
and their slopes correspond to \((\alpha \phi _{0}/8\pi )[1/\lambda _{\parallel }(T)^{2}]\) .
The fit also yields the values of \(\beta H_{\textrm {c2}\perp }\) ,
which is approximately 50 T at 2 K.
Conventionally, the \(\beta \)  value ranges from 1-2, e.g., 1.44 in the case of
\(\alpha \)  = 0.77 [2]}.
\(H_{\textrm {c2}_{\perp }}\)  of \(\kappa \) -HgBr at low temperatures is reported to be
\(\sim \) 15 T by torque measurements [6]} and largely exceeds 9 T
(the highest field studied) by the Nernst effect,
which is susceptible to vortex liquidity [7]},
suggesting an extraordinarily large \(H_{\textrm {c2}\perp }\)  for \(T_\textrm {c}\)  = 4.2 K.
Nevertheless, the \(\beta \)  value in \(\kappa \) -HgBr likely exceeds the conventional range;
probably, \(\beta \)  = 3-4. Although there is an argument on possible modifications of
Eq. (1) [8]}, considering that the \(\beta \)  in \(\kappa \) -Br takes
a conventional value approximately in the range of 1-2 [3]}, [10]},
the large \(\beta \)  value may characterize the peculiar nature of
superconductivity in \(\kappa \) -HgBr.
The temperature dependence of 1/\(\lambda _{\parallel }(T)^{2}\)  is shown in Fig. REF (b).
The 1/\(\lambda _{\parallel }(T)^{2}\)  data are fitted by the formula
\(1/\lambda _{\parallel }(T)^{2} =1/\lambda _{\parallel }(0)^{2} \times (1-(T/T_{\textrm {c}})^{p})\)  with exponent \(p\) ,
penetration depth \(\lambda _{\parallel }(0)\)  and \(T_{\textrm {c}}\)  as fitting parameters.
The gray line is the best fit with \(p\)  = 2.4, \(\lambda _{\parallel }(0)\)  = 1820 nm and
\(T_{\textrm {c}}\) = 4.0 K and, for reference, the dotted line indicates a fit with
\(p\)  fixed at the empirically adopted value, 2,
which yields \(\lambda _{\parallel }(0)\)  = 1740 nm and \(T_{\textrm {c}}\)  = 4.0 K.
The \(\lambda _{\parallel }(0)\)  value in \(\kappa \) -HgBr is three to four times larger than
those in the half-filled systems: 490-650 nm in \(\kappa \) -Br
[3]}, [12]}, [13]} and 430-535 nm in \(\kappa \) -NCS
[13]}, [15]}.
With these \(\lambda _{\parallel }(0)\) ) values,
the relation \(1/\lambda _{\parallel }^{2} = \mu _{0}n_{\textrm {s}}e^{2}/m^{\ast }\)  yields
\(n_{\textrm {s}} / m^{\ast } = 9.4 \times 10^{54}\) , \(1.0(\pm 0.3)\times 10^{56}\) , and
\(1.4(\pm 0.3)\times 10^{56}\)  (m\(^{3}\) kg)\(^{-1}\)  in
\(\kappa \) -HgBr, \(\kappa \) -Br and \(\kappa \) -NCS, respectively (Fig. REF (a)).
Further assuming that \(m^{\ast }\)  is proportional to the specific heat \(\gamma \) ,
which is 55-57 mJ/K\(^{2}\) mol for \(\kappa \) -HgBr [6]}, [17]}, 22-28 mJ/K\(^{2}\) mol
for \(\kappa \) -Br [18]}, [19]} and
25-35 mJ/K\(^{2}\) mol for \(\kappa \) -NCS [19]}, [21]},
and considering the conducting plane cross-sectional area of the unit cell,
one obtains \(n_{\textrm {s}} = 4.9 \times 10 ^{19}\) , \(2.0 (\pm 0.6)\times 10^{20}\) ,
and \(3.3 (\pm 0.9) \times 10^{20}\)  (cm)\(^{-3}\)  in \(\kappa \) -HgBr, \(\kappa \) -Br and
\(\kappa \) -NCS, respectively (Fig. REF (b)).
\(\kappa \) -Br is located very close to the Mott transition [22]},
which may be responsible for the smaller \(n_{\textrm {s}}\)  value in
\(\kappa \) -Br than in \(\kappa \) -NCS even in the half-filled case.
These values give the relative ratio of the superfluid density as
\(n_{\textrm {s}}(\kappa \) -HgBr)/\(n_{\textrm {s}}(\kappa \) -NCS) = 15\(\pm \) 4% or
\(n_{\textrm {s}}(\kappa \) -HgBr)/\(n_{\textrm {s}}(\kappa \) -Br) = 25\(\pm \) 8%.
Clearly, the superfluid density in \(\kappa \) -HgBr is significantly reduced from
that of the half-filled systems, suggesting that only a fractional spectral weight approximately
equivalent to the doped hole density contributes to the superconducting condensate.
This behavior is called gossamer superconductivity,
which is argued to emerge in doped Mott insulators.
